nLab G-∞-operads

Idea

$G$-$\infty$-operads are to (∞,1)-operads as equivariant symmetric monoidal categories are to symmetric monoidal categories

Definition

G-$\infty$-operads may be defined as fibrous patterns over the Burnside category $\mathrm{Span}(\mathbb{F}_G)$. Explicitly:

Definition

A $G$-$\infty$-operad is a functor $\pi_{\mathcal{O}}:\mathcal{O}^{\otimes} \rightarrow \mathrm{Span}(\mathbb{F}_G)$ such that

1. $\mathcal{O}^{\otimes}$ has $\pi_{\mathcal{O}}$-cocartesian lifts for backwards maps

2. For every $S \in \mathbb{F}_{\mathcal{T}}$, cocartesian transport along $\pi$-cocartesian lifts lying over the inclusions $(S \leftarrow U = U \mid U \in \mathrm{Orb}(S))$ implement an equivalence

   $$\mathcal{O}_S \simeq \prod_{U \in \mathrm{Orb}(S)} \mathcal{O}_U,$$

   where $\mathcal{O}_S = \pi^{-1}_{\mathcal{O}}(S)$.

3. For every map of orbits $T \rightarrow S$ and pair of objects $(\mathbf{C},\mathbf{D}) \in \mathcal{O}_T \times \mathcal{O}_S$, writing $\mathbf{D} = (D_U \mid U \in \mathrm{Orb}(S))$, postcomposition with the $\pi$-cartesian lifts yields an equivalence

   $$\mathrm{Map}_{\mathcal{O}^{\otimes}}^{T \rightarrow S}(\mathbf{C}, \mathbf{D}) \simeq \prod_{U \in \mathrm{Orb}(S)} \mathrm{Map}_{\mathcal{O}^{\otimes}}^{T \leftarrow T_U \rightarrow U}(\mathbf{C}, D_U),$$

   where $T_U \coloneqq T \times_S U$.

A morphism of $G$-$\infty$-operads is a functor over $\mathrm{Span}(\mathbb{F}_G)$ preserving cocartesian lifts for backwards maps.

Proposition

A functor $\mathcal{C}^{\otimes} \rightarrow \mathrm{Span}(\mathbb{F}_G)$ is simultaneously a $G$-$\infty$-operad and a cocartesian fibration if and only if it is the unstraightening of a G-symmetric monoidal $\infty$-category

Definition

If $\mathcal{O}^{\otimes},\mathcal{P}^{\otimes}$ are $G$-$\infty$-operads, an $\mathcal{O}$-algebra in $\mathcal{P}^{\otimes}$ is a morphism of $G$-$\infty$-operads $\mathcal{O}^{\otimes} \rightarrow \mathcal{P}^{\otimes}$; we denote by

$$\mathrm{Alg}_{\mathcal{O}}(\mathcal{P}) \subset \mathrm{Fun}_{/\mathrm{Span}(\mathbb{F}_G)}(\mathcal{O}^{\otimes}, \mathcal{P}^{\otimes})$$

the full subcategory spanned by $\mathcal{O}$-algebras in $\mathcal{P}$.

In particular, if $\mathcal{C}^{\otimes}$ is a $G$-symmetric monoidal $\infty$-category, then $\mathcal{O}$-algebras in $\mathcal{C}^{\otimes}$ are defined to be $\mathcal{O}$-algebras in the underlying $G$-$\infty$-operad of $\mathcal{C}^{\otimes}$.

The underlying $G$-category and $G$-symmetric sequence

Definition

Given $\mathcal{P}^{\otimes}$ a $G$-$\infty$-operad, the underlying $G$-$\infty$-category of $\mathcal{P}^{\otimes}$ is $U \mathcal{P} \coloneqq \mathcal{P}^{\otimes} \times_{\mathrm{Span}(\mathbb{F}_G)} \mathcal{O}_G^{\mathrm{op}}$.

Explicitly, the $H$-value of $U\mathcal{P}$ is the fiber $\mathcal{P}_H \coloneqq \pi_{\mathcal{P}}^{-1}([G/H])$, and the restriction functor $\Res_K^H:\mathcal{P}_H \rightarrow \mathcal{P}_K$ is pullback. We say that $\mathcal{P}^{\otimes}$ has one object if $U\mathcal{P}$ is the terminal G-∞-category.

Definition

If $\mathcal{P}^{\otimes}$ is a one-object $G$-operad, then its underlying G-symmetric sequence is the functor $\mathcal{P}(-):\tot \underline{\Sigma}_G \rightarrow \mathcal{S}$ defined by

$$\mathcal{P}(S) \coloneqq \Map_{\cP^{\otimes}}^{\Ind_H^G S \rightarrow [G/H]}(\Ind_H^G S, [G/H]).$$

This is significant largely due to the following proposition.

Proposition

A map of $G$-operads $\mathcal{P}^{\otimes} \rightarrow \mathcal{Q}^{\otimes}$ is an equivalence if and only if, for all subgroups $H \subset G$ and all finite $H$-sets $S \in \mathbb{F}_H$, the induced map $\varphi(S):\mathcal{P}(S) \rightarrow \mathcal{Q}(S)$ is an equivalence.

Examples

• N-∞ operads / G-d-operad

• EV operads

• equivariant linear isometries operad

Related concepts

• Parametrized Higher Category Theory and Higher Algebra

• equivariant symmetric monoidal category

• G-symmetric sequence

• algebraic pattern

References

Originally,

• Denis Nardin, Jay Shah, Parametrized and equivariant higher algebra, (2022) (arxiv:2203.00072)

In terms of the Burnside category,

• Shaul Barkan, Rune Haugseng, Jan Steinebrunner, Envelopes for Algebraic Patterns, (2022) (arXiv:2208.07183)

• Natalie Stewart: Equivariant operads, symmetric sequences, and Boardman-Vogt tensor products (2025) [arXiv:2501.02129]